f(x)=x^2-2x+3=(x-1)^2+2,
f(0)=3,x从0开始变大时,f(x)变小,当x=1时,f(x)有最小值2,f(x)在[0,m]上最小值为2,可见1≤m.
x从1开始变大时,f(x)也变大,当x=2时,f(x)=3,f(x)在[0,m]上最大值为3,可见m≤2.
m的取值范围是1≤m≤2.